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TRANSCRIPT
Simulation Tool for Variably Saturated Flow
with Comprehensive Geochemical Reactions in
Two and ThreeDimensional Domains
L. Wissmeier*, D.A. Barry
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Laboratoire de Technologie Ecologique
Station 2, CH‐1015 Lausanne, Switzerland
Resubmitted on 17 August 2010 to Environmental Modelling & Software
Author to whom all correspondence should be addressed. Telephone: +41 21 693 5727, facsimile: +41 21 693 8035
E‐mail addresses: [email protected] (L. Wissmeier), [email protected] (D.A. Barry)
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Abstract 1
We present a software tool for simulations of flow and multi‐component solute transport in 2
two and three‐dimensional domains in combination with comprehensive intra‐phase and 3
inter‐phase geochemistry. The software uses IPhreeqc as a reaction engine to the multi‐4
purpose, multidimensional finite element solver COMSOL Multiphysics® for flow and 5
transport simulations. Here we used COMSOL to solve Richards’ equation for aqueous phase 6
flow in variably saturated porous media. The coupling procedure presented is in principle 7
applicable to any simulation of aqueous phase flow and solute transport in COMSOL. The 8
coupling with IPhreeqc gives major advantages over COMSOL’s built‐in reaction capabilities, 9
i.e., the soil solution is speciated from its element composition according to thermodynamic 10
mass action equations with ion activity corrections. State‐of‐the‐art adsorption models such 11
as surface complexation with diffuse double layer calculations are accessible. In addition, 12
IPhreeqc provides a framework to integrate user‐defined kinetic reactions with possible 13
dependencies on solution speciation (i.e., pH, saturation indices, and ion activities), allowing 14
for modelling of microbially mediated reactions. Extensive compilations of geochemical 15
reactions and their parameterization are accessible through associated databases. 16
Keywords 17
Geochemistry, Reaction, PHREEQC, IPhreeqc, Richards’ Equation, Unsaturated Flow, 18
COMSOL, Solute Transport 19
Research highlights 20
Coupling of COMSOL and PHREEQC facilitates simulation of variably saturated flow 21
with comprehensive geochemical reactions. 22
The use of finite elements allows for the simulation of flow and solute transport in 23
complex 2 and 3D domains. 24
Geochemical reactions are coupled via sequential non‐iterative operator splitting. 25
The software tool provides novel capabilities for investigations of contaminant 26
behaviour in variably saturated porous media and agricultural management. 27
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Software requirements 28
COMSOL Multiphysics® including Earth Science Module (tested version: 3.5a; due to a 29 memory leak in versions 4.0 and 4.0a, these are not suitable for the presented coupling) 30 Price for single user academic license including Earth Science Module ca. 2000 € 31
Matlab® (tested versions: 7.9, 7.10) 32 Price for single user academic license including Parallel Computing Toolbox ca. 650 € 33
IPhreeqc (COM‐version, available free of charge at 34 http://wwwbrr.cr.usgs.gov/projects/GWC_coupled/phreeqc/) 35
The coupling files together with animations of the presented simulations are available at 36 http://infoscience.epfl.ch/record/143469 (link: download fulltext) 37
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1 Introduction 38
Computer simulations of water availability and quality play an important role in state‐of‐39
the‐art water resources management and agriculture (Barry, 1992; Šimůnek and Bradford, 40
2008). Due to the importance of the unsaturated zone as the major life‐supporting 41
ecosystem on land (Brussaard et al., 2007), there is a great demand for understanding and 42
predicting the complex interactions between highly non‐linear vadose zone flow and bio‐43
geochemical reactions (Wissmeier et al., 2009). However, many of the most utilized 44
software programs focus either on physical flow and transport phenomena (e.g., MODFLOW 45
(Harbaugh et al., 2000), SUTRA (Voss and Provost, 2008), HYDRUS (Šimůnek et al., 2009)), or 46
on geochemical reactions (e.g., MINTEQ (Gustafsson, 2010), CHESS (van der Lee, 1998), 47
ORCHESTRA (Meeussen, 2003)). Saturated flow codes have been coupled with geochemical 48
reaction packages to produce combined water flow, multispecies transport and complex 49
geochemical reactions (e.g., PHT3D (Prommer et al., 1999), DART (Freedman and Ibaraki, 50
2002), PHAST (Parkhurst et al., 2004), PHWAT (Mao et al., 2006). In recent years, simulation 51
tools for unsaturated porous media flow with detailed geochemistry have emerged (HP1 52
(Šimůnek et al., 2009), RICH‐PHREEQ (Wissmeier and Barry, 2010)). Although powerful tools, 53
the applicability of the aforementioned unsaturated flow/transport/reaction models is 54
restricted to systems with one‐dimensional movement of water and solutes. 55
The coupling of COMSOL (COMSOL Multiphysics®, 2010) and IPhreeqc (Charlton and 56
Parkhurst, 2010) presented here extends the capabilities of these models by combining 57
unsaturated porous media flow in two and three dimensions with comprehensive 58
geochemical reactions. IPhreeqc is a version of PHREEQC (Parkhurst and Appelo, 1999) that 59
is specifically designed for coupling to multicomponent transport simulators. While it retains 60
all of PHREEQC’s reaction capabilities, IPhreeqc provides additional methods for data 61
manipulation and communication to the host application. The coupled model’s main 62
features are listed below. 63
Unsaturated Flow: 64
Constitutive relations according to van Genuchten (1980), Brooks and Corey (1964) or 65
user‐defined 66
Anisotropic unsaturated conductivity 67
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Specific storage according to fluid compressibility or user‐defined storability function 68
Fluid density, viscosity and liquid source term through user‐defined expressions with 69
possible dependencies on COMSOL variables 70
Multispecies Solute Transport: 71
Dispersivity in directions parallel and orthogonal to flow 72
Tortuosity according to Millington and Quirk (1961) or user‐defined 73
Species independent coefficient of molecular diffusion and dispersivities 74
Liquid source concentrations and solute sources through user‐defined expressions with 75
possible dependencies on COMSOL variables 76
Geochemical Reactions: 77
Equilibrium solution speciation and redox reactions 78
Equilibrium mineral dissolution/precipitation 79
Ion exchange on permanently charged adsorption sites 80
Surface adsorption according to the diffuse double layer model (Dzombak and Morel, 81
1990) or the CD_MUSIC model (Hiemstra et al., 1989) with or without explicit calculation 82
of the diffuse double layer composition 83
Kinetic reactions according to user‐defined rate equations, with possible dependencies 84
on solution speciation, temperature or moisture content 85
Gas phase exchange 86
Solid solutions 87
A considerable advantage of the coupling results from the flexibility of COMSOL to define 88
any input parameter as a function of internal variables. For example, root water uptake, 89
entered as a negative liquid sources term, can be defined by a function of space coordinates 90
and liquid phase pressure head. In addition, IPhreeqc’s capabilities to integrate user‐defined 91
rate expressions allows for the simulation of complex time‐dependent geochemical 92
reactions with explicit dependencies on solution speciation (e.g., ion activities, pH, mineral 93
saturation). With these capabilities, the presented model is significantly more 94
comprehensive than existing codes in the field of unsaturated reactive transport modelling, 95
such as MIN3P (Mayer et al., 2002) and RETRASO (Saaltink et al., 2004). The model setup is 96
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greatly facilitated by graphical user interfaces for flow and transport (COMSOL) and 97
reactions (PHREEQC). 98
This paper is organized as follows. Section 2 briefly describes the underlying theory of 99
component‐based phase flow, section 3 illustrates the coupling procedure, and sections 4 100
and 5 provide application examples for kinetic pesticide degradation and fertigation via a 101
subsoil drip irrigation system (files for these cases are available for download). 102
2 Theory 103
In the presented scheme, the classical advection diffusion/dispersion equation was used for 104
multi‐component solute transport (Bear, 1972, 2007; Bear and Bachmat, 1998), 105
θ· · θ , (1)
where θ is the relative liquid phase saturation (m3 m‐3), (kg m3) is the concentration of 106
solution species , (m s‐1) is the Darcy flux and (m2 s‐1) is the hydrodynamic dispersion 107
tensor. 108
Recognizing that the liquid phase is composed of solution species according to 109
θ∑
ρ, (2)
where (mol m3) is the moles of solution species in a control volume with a molar weight 110
(kg mol‐1), the application of Eq. (1) to all solution species in the liquid phase effectively 111
yields phase mass balance according to 112
ρθ· ρθ , (3)
where is the barycentric mass flow velocity of the liquid phase (Corey and Auvermann, 113
2003). This implies that the net mass flux induced by diffusion/dispersion (Letey et al., 1969) 114
can be neglected , which is a common assumption (Jacques and Šimůnek, 2010; Miller et al., 115
1998; Šimůnek et al., 2006; Šimůnek et al., 2009). 116
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In the presented scheme, the Darcy flux θ (m s‐1) in is computed from Richards’ 117
equation according to 118
ρρ , (4)
(e.g., Bear, 1972, 2007), where is the unsaturated hydraulic conductivity tensor (m s‐1), 119
is the fluid pressure (kg m‐1 s‐2) and is the magnitude of gravitational acceleration (m s‐2). 120
The other major part of the model concerns geochemical reactions (including biological 121
reactions modelled using a geochemical approach). Thorough descriptions of the models for 122
this part are available in textbooks (Appelo and Postma, 2005; Langmuir, 1997) and the 123
PHREEQC manual (Parkhurst and Appelo, 1999). 124
3 Coupling procedure 125
In the discretized time domain of the numerical model, the three simultaneous and 126
dependent processes flow, solute transport and reaction are separated using a non‐iterative 127
sequential split‐operator approach where is the length of the coupling time step. The 128
coupling time step is further reduced within the iterative schemes for unsaturated flow, 129
solute transport and geochemical speciation as required by the respective algorithms. As 130
result of the decoupling, concentrations are assumed independent of reactions in solute 131
transport computations. As well, the liquid phase saturation and density are similarly 132
assumed independent of concentrations and reactions during flow calculations. During the 133
geochemical reaction step (performed within IPhreeqc), the solution composition is 134
assumed independent of both flow and transport. The influence of reaction kinetics, grid 135
size and splitting time step on the introduced splitting error in unsaturated flow situations 136
was investigated by Jacques et al. (2006). General discussions on operator splitting are 137
provided by Yeh and Tripathi (1989), Valocchi and Malmstead (1992) and Barry et al. (2000; 138
1996; 1997), among others. 139
Fig. 1. Program flow and structure. 140
Fig. 1 illustrates the program structure for the coupling of COMSOL and IPhreeqc using 141
Matlab (Matlab®, 2010) as the controlling application and interface for the COMSOL and 142
IPhreeqc modules (grey shaded, dashed boxes). The code is organized in three major m‐files 143
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(black textboxes in Fig. 1). At the beginning of the simulation, the controlling script 144
coupling_COMSOL_PHREEQC.m runs subscripts to configure the simulation domain with 145
flow and transport properties, initializes COMSOL and IPhreeqc computations and specifies 146
boundary conditions. Geochemical properties such as pH and redox potential (pe) are 147
extremely sensitive to the total element concentrations. Minimal deviations of the ratio of 148
oxygen to hydrogen from the stoichiometric ratio of 1:2 in the simulation of pure water will 149
dramatically affect the pe, for example. Precise concentration values for initial and 150
boundary conditions in COMSOL are thus conveniently determined by preliminary PHREEQC 151
speciation calculations. 152
After the initialization procedure, computations enter the iteration loop of the split 153
operator, where COMSOL and IPhreeqc procedures alternate for the calculation of flow, 154
transport and reactions. Data from the output of one procedure is passed back to the 155
Matlab workspace and reformatted for the following procedure using the functions 156
phreeqc2comsol and comsol2phreeqc. For the computation of aqueous phase flow in 157
COMSOL, fluid pressures are calculated from the output of the preceding reaction step 158
(mass of solution in control volume, solution density, pressure‐saturation relation). Darcy 159
flux velocities from flow calculations together with master species concentrations from the 160
reaction step are input into the COMSOL solute transport routine (Eq. (1)). The use of 161
master species instead of solution species considerably reduces the degrees of freedom in 162
transport calculations and therefore the numerical burden (Steefel and MacQuarrie, 1996). 163
However, it impedes the simulation of species‐dependent diffusion. The comsol2phreeqc‐164
function retrieves the results from COMSOL as a vector of all degrees of freedom (fluid 165
pressure and element concentrations) at all nodes in the domain. The vector is reformatted 166
as a numeric Matlab array. Master species concentrations from transport calculations are 167
multiplied by moisture contents from flow calculations. This yields the aqueous phase 168
composition at each node in terms of moles of master species per control volume. In the 169
IPhreeqc reaction step, each node in the simulation domain is represented by a single batch 170
reactor that contains physical amounts (moles) of solution elements, minerals, gases and 171
adsorption sites. Since only the liquid phase composition is altered through flow and 172
transport, comsol2phreeqc only updates the solution composition using IPhreeqc's method 173
SOLUTION_MODIFY. From the ensemble of elements, the aqueous solution is speciated, and 174
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mineral reactions, surface adsorption/desorption reaction and kinetic reactions are 175
computed. After the IPhreeqc reaction procedure, flow and transport calculations in 176
COMSOL are reinitialized with the updated values of moisture content and master species 177
concentrations using the phreeqc2comsol‐function. Results, which are passed back from 178
IPhreeqc to the Matlab workspace as a cell array, are reshaped as a vector with the same 179
structure of the COMSOL solution vector. COMSOL’s asseminit‐function is used to create 180
valid initial conditions for flow and transport calculations in COMSOL. 181
All primary master species in the liquid phase (including elemental oxygen and hydrogen) 182
must be included in solute transport calculations in order to represent flow by the transport 183
of liquid phase components. This approach has been applied in previous couplings of 184
geochemistry in PHREEQC with unsaturated flow and transport calculations (Jacques et al., 185
2008; Wissmeier and Barry, 2010). Charge imbalance, present in initial and boundary 186
solutions or developing during reactions (e.g., surface adsorption without explicit calculation 187
of the diffuse layer composition) (Parkhurst, 2010), is treated as an extra solution 188
component in transport calculations. It is also included in speciation calculations in order to 189
accurately reproduce pH and pe. 190
During the conversion from units relative to an arbitrary control volume in COMSOL (e.g., 191
m3 m‐3 for θ) to physical quantities in IPhreeqc, the size of the batch reactors is taken as one 192
litre for convenience. Thus, solid phase properties in IPhreeqc, such as the moles of 193
adsorption sites and mineral phases have to be understood as per litre of unsaturated zone 194
volume (soil including pore space). Without further modifications, solid phase properties are 195
independent of water contents, i.e., the moles of mineral phases or exchange sites in 196
contact with the unsaturated soil solution do not change due to changes in moisture 197
contents. 198
Negative master species concentrations may result from oscillatory behaviour of COMSOL’s 199
finite element scheme for solute transport in the vicinity of sharp concentration gradients. 200
The performance index, defined as the product of the grid Péclet number and the Courant 201
number, can be used as an indicator of oscillatory transport behaviour (Huyakorn and 202
Pinder, 1983; Perrochet and Berod, 1993; Šimůnek et al., 2009). Since IPhreeqc defines the 203
solution composition in terms of moles of elements, negative concentrations from the 204
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COMSOL‐procedure are set to zero during reaction calculations. After the reaction step, 205
negative concentrations from the previous flow/transport step in COMSOL are summed to 206
the master species concentrations that are output of IPhreeqc. The summation of negative 207
concentrations from transport simulations to the results from reaction calculations avoids 208
significant mass balance errors that would otherwise result from the repeated adjustment 209
of negative output from COMSOL to zero. This procedure remedies mass balance errors. 210
Nevertheless, it is recommended to avoid oscillations in solute transport through 211
appropriate domain discretisation (Donea and Huerta, 2004; Perrochet and Berod, 1993). 212
For accurate transport of pH in unbuffered solutions, the relative tolerance of the solute 213
transport scheme in COMSOL is set to 10‐10. In order to decrease the absolute tolerance for 214
oxygen and hydrogen, their nominal concentrations in COMSOL are reduced by their molal 215
concentrations in pure water (55.50621679636 molO/kgwater and 111.0124335927 216
molH/kgwater, respectively). Even with this high accuracy, calculations of the redox potential 217
(pe) in unbuffered solutions may not give reliable results and should be examined carefully. 218
Equilibration with atmospheric O2 stabilizes otherwise unbuffered solutions und should 219
therefore be considered if applicable (Parkhurst and Appelo, 1999). 220
The COM‐version of IPhreeqc (Charlton and Parkhurst, 2010) that is utilized in the presented 221
coupling has a number of features that greatly simplify data handling: 222
Reaction calculations can be entirely controlled from within Matlab by passing 223
commands strings to the COM object. 224
Changes in the reaction system due to transport of aqueous elements are considered by 225
modifying the moles of elements in solution. 226
No cumbersome writing/reading of files is necessary in order to pass results from the 227
reaction module to the flow/transport module and vice versa. Data transfer via the 228
Matlab workspace is processed entirely in memory. 229
The status of the entire geochemical system, including all exchangers, surfaces, kinetics 230
phases, mineral and gas phases, is automatically saved by the COM object in between 231
calls. 232
As result of persistence of the geochemical properties in the COM object, the coupling 233
procedure only needs to manage solution elements. This not only decreases programming 234
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effort and increases coupling efficiency but also eliminates the need for user‐defined 235
definitions of all possible exchange and surface species, as is found in earlier couplings 236
involving PHREEQC (Prommer et al., 2003). 237
4 Verification example: Pesticide degradation 238
This application example reproduces the simulation of kinetic pesticide degradation in 239
COMSOL’s model library (COMSOL Multiphysics®, 2008b) using IPhreeqc instead of 240
COMSOL’s Reaction Engineer Lab® to simulate the reaction chain. According to 241
COMSOL Multiphysics® (2008b), the simulation is inspired by but does not exactly duplicate 242
the application example no. 7 of the software packages SWMS‐2D (Šimůnek et al., 1994) and 243
HYDRUS‐2D (Šimůnek et al., 1999). Because of the identical treatment of unsaturated flow 244
and solute transport, the application example can be regarded as a verification of the 245
coupling procedure. 246
Fig. 2. Simulation domain with finite element mesh for pesticide simulation. The source zone is indicated by the thick 247 line at y = 0. 248
The radially symmetric simulation domain with 4401 nodes and 26406 degrees of freedom is 249
displayed in Fig. 2. The pesticide Aldicarb enters the dry soil via constant flux infiltration at 250
the top boundary in the region y = 0 m, 0 m ≤ r ≤ 0.2 m. The soil is divided into two 251
subdomains with different hydraulic properties, i.e., higher conductivity and lower saturated 252
water content below y = ‐0.4 m. The initial hydraulic head (m) of the simulation domain is 253
given by the expression 254
1.2 m , 0.4 ,1.2 1.6 m , 0.4 .
(5)
Transformations between Aldicarb and its breakdown products are governed by first‐order 255
rates according to Eqs. (6‐11), where denotes moles of the chemical in the control 256
volume (mol m‐3), are reaction constants (s‐1), denote total molal concentrations 257
(mol kgwater‐1) and stands for the kilograms of water in the control volume (kg m‐3). 258
Aldicarb (a): aa a ; (6)
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Aldicarb Oxime (ao): aoa ; (7)
Aldicarb Sulfoxide (asx): asxa asx asx ; (8)
Aldicarb Sulfoxide Oxime (asxo): asxoasx ; (9)
Aldicarb Sulfone (asn): asnasn asn ; (10)
Aldicarb Sulfone Oxime (asno): asnoasn . (11)
The reaction chain together with first‐order rate constants is displayed schematically in Fig. 259
3. 260
Fig. 3. Aldicarb reaction chain. 261
To be consistent with the COMSOL example, it is necessary to prevent unwanted 262
geochemical transformations. Therefore, Aldicarb and its daughter products were 263
introduced into IPhreeqc as separate uncharged solution master species with no reactions 264
apart from the transformations listed in Fig. 3. Further details of the breakdown process and 265
the COMSOL solution are available in the COMSOL model library documentation 266
(COMSOL Multiphysics®, 2008a). The coupling time step Δ was set to 0.1 d. 267
Fig. 4. Comparison of concentrations from the coupling procedure and COMSOL alone eight d after beginning of 268 infiltration. 269
In order to verify the coupling procedure, moisture content and concentrations were 270
compared along the centre line (r = 0) with results from COMSOL alone (Fig. 4). Moisture 271
contents from the coupled simulation (COMSOL+IPhreeqc) and the global implicit method in 272
COMSOL (COMSOL) are visually indistinguishable. Small differences in chemical 273
concentrations can be fully attributed to the splitting error, whose magnitude decreases 274
linearly with Δ . If negative concentrations from the previous transport step are not 275
summed to the concentrations after transport, as described in section 3, an obvious mass 276
balance error with overestimation of all pesticide species is produced. Thus, the results 277
prove the accuracy of the coupling procedure and particularly the successful suppression of 278
significant mass balance errors that would result from the uncompensated adjustment of 279
concentrations to positive values during reaction calculations. 280
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5 Application example: Drip irrigation and fertigation 281
This example, reported by Hanson et al. (2006), simulates the fertigation of a tomato 282
plantation using a subsoil micro‐irrigation system with periodic input of water and fertilizer 283
solution. The simulation of flow and transport was reproduced in COMSOL using the same 284
domain properties and parameters as in Hanson et al. (2006), who applied HYDRUS‐2D 285
(Šimůnek et al., 1999). In contrast to Hanson et al. (2006), however, we implemented a 286
detailed description of geochemical through the coupling to IPhreeqc. This included solution 287
speciation with ion activity correction, kinetic redox transformations and cation exchange. 288
In agreement with Hanson et al. (2006), root water uptake was simulated using the spatial 289
root distribution model of Vrugt et al. (2001) in combination with the water stress response 290
function of Feddes et al. (1978). The model was implemented through the Matlab function 291
root_uptake.m, which is called from COMSOL at runtime (details on root water uptake are 292
given by Šimůnek and Hopmans, 2009; Šimůnek et al., 1999). 293
Fig. 5. Simulation domain with finite element mesh for drip irrigation simulation. 294
The simulation domain with 2665 nodes and 23985 degrees of freedom is displayed in Fig. 5. 295
It represents a 2D slice of the soil system with axial symmetry along the left boundary. 296
Solutes and irrigation water enter the soil through the small half circle at x = 0 m, y = ‐0.2 m 297
that represents the irrigation pipe with a diameter of 0.02 m. Irrigation imposes a constant 298
flux for 1.15 d intermittent with no flow periods of 2.35 d. The fertilizer enters the domain 299
for 0.575 d via a constant concentration boundary beginning 0.2875 d after start of 300
irrigation periods. Details of the flow and solute transport properties are given by Hanson et 301
al. (2006) and Gärdenäs et al. (2005) (simulation SUBTAPE, fertigation scenario M50). 302
Master species concentrations in the inflowing fertilizer solution were calculated by reacting 303
0.1449 mol KH2PO4, 0.0362 mol NH4NO3 [AmmHNO3] and 0.0725 mol (NH2)2CO [Urea] with 304
1 kg of pure water. In order to prevent instantaneous hydrolysis and nitrification, urea 305
(Urea) and ammonia (Amm) were defined as separate solution master species in IPhreeqc. 306
In addition, the fertilizer solution and the fertilizer‐free irrigation water were equilibrated 307
with atmospheric CO2 (10‐3.5atm) and O2 (10
‐0.7 atm). The initial soil solution was assumed in 308
equilibrium with atmospheric O2 (10‐0.7 atm) and with CO2 from root respiration at a partial 309
pressure 10‐1.5 atm. The latter value was chosen in agreement with the expected high actual 310
evaporation (~1200 mm yr‐1) for an irrigated tomato plantation in California, USA (Hanson et 311
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al., 2006) according to Brook et al. (1983). Properties of inflowing and initial solutions are 312
summarized in Table 1. 313
Table 1 314 Geochemical solution properties. Master species concentrations in mol kgwater
‐1, truncated to 2 digits. 315
Equilibrium solution speciation was defined as given in the thermodynamic database 316
phreeqc.dat, which is distributed with IPhreeqc. In order to suppress denitrification as in 317
Hanson et al. (2006), the equilibrium constant for the reaction 2NO 12H 10e N318
6H O was set to 10‐1000. Kinetic hydrolysis of urea was simulated according to the reaction 319
Urea H O 2Amm CO , (12)
with the first‐order rate expression (mol s‐1 m‐3) (Hanson et al., 2006), 320
hyd hyd Urea , (13)
where hyd is the reaction constant taken as ‐0.35 d‐1 and Urea is the molal urea 321
concentration. For the kinetic nitrification of ammonium to nitrate, we used the reaction 322
formula 323
AmmH 2O NO 2H H O, (14)
with the rate, 324
nit nit Amm O 0
O 0 . (15)
The reaction constant nit was set to ‐0.2 d‐1. In Eq. (15), Amm and O 0 are the molal 325
concentrations of ammonium and zero‐valent oxygen, respectively, and O 0 is the 326
concentration of zero‐valent oxygen in the initial solution (5.11 × 10‐4 mol kgwater‐1). The last 327
term in Eq. (15) was introduced as an additional factor to the rate expression of Hanson et 328
al. (2006). Its purpose is to simulate free oxygen as a rate‐limiting redox partner for the 329
kinetic oxidation of ammonium to nitrate. 330
In contrast to Hanson et al. (2006), who used a simple linear isotherm for ammonium 331
adsorption, we simulated adsorption of aqueous cations on negatively charged surfaces as a 332
cation exchange process (Appelo and Postma, 2005). Besides potassium and ammonium, 333
15
free protons (H+) are included in the competition for the exchange sites in order to consider 334
the pH dependence of cation exchange and to be consistent with the conceptualization of 335
plant nutrient uptake via proton exchange (Hedrich and Schroeder, 1989). The cation 336
exchange capacity was taken as 150 meq lsoil‐1, which is a reasonable value for loam 337
(Yukselen and Kaya, 2006). Exchange reaction formulas and ion affinities with suggested 338
values for proton exchange were those included in the database phreeqc.dat. The split‐339
operator time step Δ for the alternation between transport and reactions calculations was 340
set to 0.01 d. 341
Fig. 6. Normalized root distribution according to the Vrugt model (Vrugt et al., 2001) with parameters from Hanson et al. 342 (2006). 343
Fig. 6 displays the normalized root distribution function used to describe water and solute 344
losses due to plant water uptake. As required by the Vrugt model (Vrugt et al., 2001), the 345
normalized root distribution integrates to unity over the simulation domain. Clearly visible 346
in Fig. 7 is the decrease of root density at depths y < ‐0.2 m, which is determined by the 347
parameter in Hanson et al. (2006). 348
Fig. 7. Moisture contents at 0 and 1.15 d, i.e., the beginning and end of the first irrigation period; white contours: lines of 349 equal water contents. 350
The successful re‐simulation of moisture dynamics was verified by visually comparing 351
moisture contents at the beginning and end of the first irrigation period with results 352
presented in Gärdenäs et al. (2005) and Hanson et al. (2006) (To follow this comparison the 353
reader is referred to the original references). The comparison reveals small discrepancies in 354
the dry part of the domain and along the right no‐flow boundary, which can be attributed to 355
different domain discretizations and tolerance settings in the numerical methods. 356
Fig. 8. Solution master species concentrations 3.88, 4.63, 5, 7 and 28 d after beginning of fertigation cycles. 357
Molal concentrations of all solution master species, except for oxygen, hydrogen and 358
carbon, are displayed in Fig. 8. In the model phosphorus (P) is not involved in any kinetic 359
reactions or adsorption/desorption processes (since it does not adsorb to negatively 360
charged sorption sites) and therefore can be used as a reference for conservative solute 361
transport. By comparison to phosphorus and from concentrations of O(0) in Fig. 9 it is 362
evident that the kinetic transformation of ammonium to nitrate is limited by the available 363
O(0). Urea is clearly affected by hydrolysis with decreasing concentrations after fertigation 364
periods and increasing concentrations of ammonia (Amm). Potassium (K) is strongly 365
16
retarded compared to phosphorus, with equal inflowing and initial concentrations due to 366
exchange with adsorbed protons and ammonia. 367
Fig. 9. Moisture contents, pH and exchanger composition 3.88, 4.63, 5, 7 and 28 d after beginning of fertigation cycles. 368
The exchanger composition, pH and molal concentrations of O(0) are presented in Fig. 9. 369
The soil’s low initial pH originates from respiratory CO2. The decreasing pH behind the solute 370
front can be attributed to (i) oxidation of ammonium and (ii) the replacement of protons 371
from the surface exchange sites by ammonia and potassium. The latter can be interpreted 372
as a proton “snow plough” (Barry et al., 1983). 373
Low pH values due to nitrification and proton exchange may result in significant dissolution 374
of soil matrix minerals (e.g., calcite), which may act as pH buffers and release cations that 375
compete for adsorption sites and thereby reduce the adsorption efficiency of fertilizer ions. 376
Kinetic or equilibrium mineral reactions can be easily included in the simulation. However, 377
this was not taken into account in the given example because of the unknown mineral 378
composition of the substrate. Kinetic oxidation of ammonium to nitrate leads to a local 379
depletion in O(0) concentrations that inhibits further nitrification according to Eq. (15). Due 380
to the absence of other cations, the initial exchanger is entirely filled with H+. This situation 381
is unlikely to be valid in the field, but allows for more straightforward interpretation of this 382
illustration. A more realistic exchanger initialization requires measured exchanger 383
compositions or knowledge about the trace elements in the initial soil solution. Even though 384
the exchanger has a slightly larger affinity for potassium, ammonium dominates the 385
exchanger composition in the exterior of the solute ring that forms around the irrigation 386
pipe. This is due to the elevated concentration of ammonium from urea hydrolysis and 387
retardation of potassium, which shifts the concentration ratio towards ammonium. In 388
addition, the low pH values in this region favour the formation of ammonium as opposed to 389
non‐adsorbing ammonia. Potassium is mainly adsorbed close to the irrigation pipe, where 390
ammonium concentrations are low. The region at x = 0.6 m, y = 0 m remains extremely dry 391
(cf. Fig. 7). Therefore, it is effectively inert towards changes in geochemical conditions (e.g., 392
pH, O(0) content). 393
Results of the present simulation and those of Hanson et al. (2006) show roughly similar 394
patterns for nitrate and urea. However, in our simulations nitrate concentrations at 7 and 28 395
17
d are considerably lower in the direct vicinity of the supply tube, where it shows the highest 396
concentrations in Hanson et al. (2006). The lower concentration in our simulations is 397
expected considering the irrigation/fertigation scheme where the fertilizer is flushed into 398
the soil with pure water following its application. 399
6 Concluding remarks 400
The application example in section 4 verifies the coupling of unsaturated flow and 401
multispecies solute transport in COMSOL with the geochemical modelling framework 402
IPhreeqc. Minor differences to the global implicit solution are introduced through operator 403
splitting and can be controlled by the magnitude of the splitting time step Δ . Furthermore, 404
it was shown that the constraint of positive concentrations in IPhreeqc in combination with 405
negative concentrations from COMSOL’s solute transport scheme does not lead to 406
significant mass balance errors. 407
Section 5 shows the enhanced geochemical capabilities resulting from the coupling between 408
COMSOL and IPhreeqc in a complex simulation of unsaturated flow and solute transport. 409
The successful replication of the moisture content distribution from Hanson et al. (2006) 410
illustrates the capabilities of COMSOL’s Earth Science Module to simulate unsaturated flow 411
processes. Further, the implementation of pressure‐ and root distribution‐dependent root‐412
water uptake demonstrates the convenience of COMSOL to extend predefined vadose zone 413
processes through user‐defined Matlab functions. 414
Due to the capabilities of COMSOL to simulate a wide variety of environmental flow 415
phenomena (e.g., two‐phase flow, fracture flow, flow according to the Darcy, Brinkman or 416
incompressible Navier‐Stokes equations), the presented coupling procedure can be utilized 417
as a general guide for linking flow and transport calculations in COMSOL to geochemical 418
reactions in IPhreeqc. 419
Acknowledgments 420
The authors wish to thank David Parkhurst and Tony Appelo for their assistance with the 421
coupling procedure and the prompt implementation of the required features into PHREEQC. 422
18
This research was supported by Alcoa World Alumina Australia. 423
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588
Table
Table 1
Geochemical solution properties. Master species concentrations in mol(kg water)‐1,
truncated to 2 digits.
Fertilizer solution Irrigation water Initial soil solution
pH 3.844 5.659 4.660 pe 16.78 14.96 15.96 O 56.12 55.51 55.51H 111.34 111.01 111.02 P 0.15 0.00 0.00 K 0.15 0.00 0.00Amm 3.62 × 10‐2 0.00 0.00 Urea 7.25 × 10‐2 0.00 0.00N 3.62 × 10‐2 0.00 0.00 C 1.04 × 10‐5 1.30 × 10‐5 1.10 × 10‐3
Figure Captions
Fig. 1. Program flow and structure.
Fig. 2. Simulation domain with finite element mesh for pesticide simulation. The source zone
is indicated by the thick line at y = 0.
Fig. 3. Aldicarb reaction chain.
Fig. 4. Comparison of concentrations from the coupling procedure and COMSOL alone eight
d after beginning of infiltration.
Fig. 5. Simulation domain with finite element mesh for drip irrigation simulation.
Fig. 6. Normalized root distribution according to the Vrugt model (Vrugt et al., 2001) with
parameters from Hanson et al. (2006).
Fig. 7. Moisture contents at 0 and 1.15 d, i.e., the beginning and end of the first irrigation
period; white contours: lines of equal water contents.
Fig. 8. Solution master species concentrations 3.88, 4.63, 5, 7 and 28 d after beginning of
fertigation cycles.
Fig. 9. Moisture contents, pH and exchanger composition 3.88, 4.63, 5, 7 and 28 d after
beginning of fertigation cycles.
comsol2phreeqc.m
phreeqc2comsol.m
Split Operator Time Step Δt
Master Species ConcentrationsMass of Water
ReactionsIntra‐Phase Speciation Inter‐Phase Speciation
Kinetic Reactions
IPhreeqc (COM version)
coupling_COMSOL_PHREEQC.m
Flow VelocitiesMoisture Content
Aqueous Phase Flow
Master Species ConcentrationsMoisture Content
Solute Transport
COMSOL
Domain GeometryDomain Discretisation (Meshing) Flow and Transport Properties Boundary and Initial Conditions PHRREQC Reaction Properties
Post‐processing
Matlab
Aldicarb
Aldicarb sulfoxide
Aldicarb sulfone
Aldicarb oxime
Aldicarb sulfoxide oxime
Aldicarb sulfone oxime
k1 = 0.2 d‐1
k2 = 0.36 d‐1
k4 = 0.024 d‐1
k5 = 0.0524 d‐1
k3 = 0.01 d‐1
x (m)
y (m
)
0 0.2 0.4 0.6-1
-0.8
-0.6
-0.4
-0.2
0
0
1
2
3
4
5
6
